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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.22190 |
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| _version_ | 1866915417136562176 |
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| author | Addas-Zanata, Salvador |
| author_facet | Addas-Zanata, Salvador |
| contents | We consider twist diffeomorphisms of the torus, $f:{\rm T^2\rightarrow T^2,}$ and their vertical rotation intervals $ρ_V(\widehat{f})=[ρ_V^{-},ρ_V^{+}],$ where $\widehat{f}$ is a lift of $f$ to the vertical annulus or cylinder. We show that $C^r$-generically for any $r\geq 1$, both extremes of the rotation interval are rational and locally constant under $C^0$-perturbations of the map. Moreover, when $f$ is area-preserving, $C^r$-generically $ρ_V^{-}<ρ_V^{+}.$
Also, for any twist map $f$, $\widehat{f}$ a lift of $f$ to the cylinder, if $ρ_V^{-}<ρ_V^{+}=p/q$, then there are two possibilities: either $\widehat{f}^q(\bullet)-(0,p)$ maps a simple essential loop into the connected component of its complement which is below the loop, or it satisfies the Curve Intersection Property. In the first case, $ρ_V^{+} \leq p/q$ in a $C^0$-neighborhood of $f,$ and in the second case, we show that $ρ_V^{+}(\widehat{f}+(0,t))>p/q$ for all $t>0$ (that is, the rotation interval is ready to grow). Finally, in the $C^r$-generic case, assuming that $ρ_V^{-}<ρ_V^{+}=p/q,$ we present some consequences of the existence of the free loop for $\widehat{f}^q(\bullet)-(0,p)$, related to the description and shape of the attractor-reppeler pair that exists in the annulus. The case of a $C^r$-generic transitive twist diffeomorphism (if such a thing exists) is also investigated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $C^r$-generic twist maps of ${\rm T^2}$ Addas-Zanata, Salvador Dynamical Systems We consider twist diffeomorphisms of the torus, $f:{\rm T^2\rightarrow T^2,}$ and their vertical rotation intervals $ρ_V(\widehat{f})=[ρ_V^{-},ρ_V^{+}],$ where $\widehat{f}$ is a lift of $f$ to the vertical annulus or cylinder. We show that $C^r$-generically for any $r\geq 1$, both extremes of the rotation interval are rational and locally constant under $C^0$-perturbations of the map. Moreover, when $f$ is area-preserving, $C^r$-generically $ρ_V^{-}<ρ_V^{+}.$ Also, for any twist map $f$, $\widehat{f}$ a lift of $f$ to the cylinder, if $ρ_V^{-}<ρ_V^{+}=p/q$, then there are two possibilities: either $\widehat{f}^q(\bullet)-(0,p)$ maps a simple essential loop into the connected component of its complement which is below the loop, or it satisfies the Curve Intersection Property. In the first case, $ρ_V^{+} \leq p/q$ in a $C^0$-neighborhood of $f,$ and in the second case, we show that $ρ_V^{+}(\widehat{f}+(0,t))>p/q$ for all $t>0$ (that is, the rotation interval is ready to grow). Finally, in the $C^r$-generic case, assuming that $ρ_V^{-}<ρ_V^{+}=p/q,$ we present some consequences of the existence of the free loop for $\widehat{f}^q(\bullet)-(0,p)$, related to the description and shape of the attractor-reppeler pair that exists in the annulus. The case of a $C^r$-generic transitive twist diffeomorphism (if such a thing exists) is also investigated. |
| title | On $C^r$-generic twist maps of ${\rm T^2}$ |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2507.22190 |